Question: The lifespans of sloths in a particular zoo are normally distributed. The average sloth lives $16.6$ years; the standard deviation is $1.1$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a sloth living between $15.5$ and $19.9$ years.
Explanation: $16.6$ $15.5$ $17.7$ $14.4$ $18.8$ $13.3$ $19.9$ $99.7\%$ $68\%$ $15.85\%$ $15.85\%$ We know the lifespans are normally distributed with an average lifespan of $16.6$ years. We know the standard deviation is $1.1$ years, so one standard deviation below the mean is $15.5$ years and one standard deviation above the mean is $17.7$ years. Two standard deviations below the mean is $14.4$ years and two standard deviations above the mean is $18.8$ years. Three standard deviations below the mean is $13.3$ years and three standard deviations above the mean is $19.9$ years. We are interested in the probability of a sloth living between $15.5$ and $19.9$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the sloths will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $68\%$ of the sloths will have lifespans within 1 standard deviation of the mean. The probability of a particular sloth living between $15.5$ and $19.9$ years is ${68\%} + \color{orange}{15.85\%}$, or $83.85\%$.